
We note that the norm constraint here has dimension \(dc\)  and \(\mathbf {vec}(\mathbf {W}_1)\)  has dimension \(s^2\) , so by [1]} this strong duality result from Lemma REF  requires that \(m^* \le \min \lbrace s^2, dc\rbrace \) . The Lagrangian is given by
\(p_{MM}^* = \max _{\mathbf {V}_i} \min _{\lambda \ge 0} \min _{\Vert \mathbf {Z}\Vert _*\le 1} -\sum _{i=1}^n \mathcal {L}^*(\mathbf {V}_i, \mathbf {Y}_i) + \sum _{j=1}^m \lambda _j \left(\beta - \sum _{i=1}^n\mathrm {trace}\left( (\mathbf {I}_d \otimes \mathbf {V}_i^T)({\mathbf {X}}_i^\top \otimes \mathbf {I}_{s}) \mathbf {Z} \right)\right)\) 
